To find which choice shows a relationship that is different from the other three choices, we need to compare the values in the table with the expression for [tex]\( b(x) = 2 \cdot 3^x \)[/tex].
Given the function:
[tex]\[ b(x) = 2 \cdot 3^x \][/tex]
Let's calculate the number of bacteria at each time point [tex]\( x \)[/tex] to see if the provided table matches this function:
1. For [tex]\( x = 0 \)[/tex]:
[tex]\[ b(0) = 2 \cdot 3^0 = 2 \cdot 1 = 2 \][/tex]
2. For [tex]\( x = 1 \)[/tex]:
[tex]\[ b(1) = 2 \cdot 3^1 = 2 \cdot 3 = 6 \][/tex]
3. For [tex]\( x = 2 \)[/tex]:
[tex]\[ b(2) = 2 \cdot 3^2 = 2 \cdot 9 = 18 \][/tex]
So, the calculated values are:
[tex]\[
\begin{array}{|c|c|}
\hline
x & b(x) \\
\hline
0 & 2 \\
\hline
1 & 6 \\
\hline
2 & 18 \\
\hline
\end{array}
\][/tex]
Now, let's compare these values with the given table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & b(x) \\
\hline
0 & 3 \\
\hline
1 & 6 \\
\hline
2 & 12 \\
\hline
\end{array}
\][/tex]
It is clear that the given table does not match our calculated values for [tex]\( x = 0 \)[/tex] and [tex]\( x = 2 \)[/tex]. The correct values should be:
- At [tex]\( x = 0 \)[/tex], the value should be 2, but the given table shows 3.
- At [tex]\( x = 2 \)[/tex], the value should be 18, but the given table shows 12.
Therefore, the relationship shown by the given table is different from the correct exponential relationship described by the formula [tex]\( b(x) = 2 \cdot 3^x \)[/tex].
